Definitions

A function consists of
- the domain set
- the codomain set
- a rule that maps members of the domain to members of the codomain in such a way that each member of the domain is mapped to at most one member of the codomain
We write $f: A\to B$ to denote the function $f$ with domain $A$ and codomain $B$. The function is total if each member of $A$ has exactly one member of the codomain associated with it; it is called partial otherwise.
If element $a\in A$ is mapped under $f$ to the element $b\in B$, we say that $a$ is the argument of $f$, and $b$ is the value of $f$, and we write $b = f(a)$.
The set $S$ ::= $\{a \in A: f(a) \text{ is defined}\}$ is called the support of the function $f$. The support is not necessarily equal to the domain.
The $R$ ::= $\{ b\in B: \text{there exists some $a\in A$ such that } f(a) = b \}$ is called the range of the function $f$. The range is not necessarily equal to the codomain.

Definitions (continued)

The domain or codomain of a function can be finite or infinite.
In case the domain is finite, the values of the function can be given as a table. E.g., let $\mathcal{T}$ be the set of truth values {T, F}. Then $\neg: \mathcal{T}\to\mathcal{T}$ is a function defined by the table
```
+-----+--------+
|  P  | NOT(P) |
+=====+========+
|  T  |  F     |
+-----+--------+
|  F  |  T     |
+-----+--------+
```
The function can also be given by a formula. E.g. Let $f : \mathbb{R}\to\mathbb{R}$ be a function given by the formula $f(x)=x^2$ for all $x\in\mathbb{R}$. This is the familiar square function we studied in calculus.

Let $f : A\to B$ be any function. There’s a natural extension of $f$ to another function $\hat{f} : \mathcal{P}(A)\to\mathcal{P}(B)$ by declaring $\hat{f}(X) = \{ f(x) : x\in X \}$ for all $X\subseteq A$.
This extension is so natural that people often blur the distinction between $f$ and $\hat{f}$ and simply write $f$ for both functions. From now on we’ll also do that.
For any $X\subseteq A$, we say that $f(X)$ is the image of $X$ under $f$.
We also define the inverse function of $f$, written $f^{-1} : \mathcal{P}(B)\to\mathcal{P}(A)$ to be $f^{-1}(Y) = \{ x\in A: f(x)\in Y\}$ for any $Y\subseteq B$. We then say that $f^{-1}(Y)$ is the preimage of $Y$ under $f$.
For $b\in B$ we also write $f^{-1}(b)$ to mean $f^{-1}(\{b\})$.

In calculus we studied the sum, difference, product, and quotient of real-valued or complex-valued functions.
A more general operation on functions called composition can be defined like this. Let $f:A\to B$ and $g: B\to C$ be functions. Then we can define $g\circ f$ to be the composition of $g$ with $f$, written $g\circ f : A\to C$, to be given by the rule $(g\circ f)(x) = g(f(x))$ for any $x\in A$.
E.g. the function $\sin(x^2):\mathbb{R}\to\mathbb{R}$ is the composition of $f: \mathbb{R}\to\mathbb{R}$ and $g: \mathbb{R}\to\mathbb{R}$ where $f(x)=x^2$ for all $x\in\mathbb{R}$ and $g(x) = \sin(x)$ for all $x\in\mathbb{R}$.
Let $f: A\to A$ be a function into itself, and let $n$ be a positive integer. We define $f^1$ to be $f$, and for $n>1$ define $f^n = f\circ f^{n-1}$.